Introduction to Classical Molecular Dynamics. Giovanni Chillemi HPC department, CINECA

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1 Introduction to Classical Molecular Dynamics Giovanni Chillemi HPC department, CINECA

2 MD ingredients Coordinates Velocities Force field Topology MD Trajectories Input parameters Analysis

3 Equation of motion The equations that describe the temporal evolution of a physical system is called equation of motion. There are different equations of motions, which characterize the motion with different levels of approximation: Time-dependent Schrödinger's Equation for quantum-mechanical system Newton s Equation for classical-mechanical system Langevin's Equation for stochastic system

4 Newton s Equation of motion Molecules are quantum-mechanical systems whose motion should be described by Schrödinger's Equation. However, technical difficulties make solving Schrödinger's Equation for large systems impractical. Therefore the motion of a molecule is usually approximated by the laws of classical mechanics and by Newton's equation of Motion. In its most simplistic form Newton's second law of motion states: fi = m i ai where mi is the mass of particle i, ai is its acceleration. The force fi is given as the derivative of the potential energy function V: where ri is the position of particle i V fi = ri

5 Potential energy function 2 2 V r, r2,..., rn kbn bn b0 n k n n 0 n bond 2 angle 2 2 k k n cos mn n n n n 0n improper 2 dihedral dihedral Cij 2 Cij rij rij nonbonded pairs ij qi q j + 4 r 0 r ij bonded interactions non bonded interactions

6 Model Degree of freedom Example of predicted properties Considered Removed Quantum mechanic Nucleus, electrons Nucleons Chemistry reaction Polarizable atoms Atoms, dipoles Electrons Binding of charged substrates Non polarizable atoms Solute atoms, solvent atoms Dipoles Conformational transitions Hydration Implicit solvent Solute atoms Solvent atoms Folding topology of macromolecules Classical Molecular Dynamics

7 A Brief History Year System 964 Liquid Argon (Rahman Phys Rev) 974 Water (Rahman J. Chem Phys. ) 977 Small protein in vacuo (Mc Cammon Karplus Nature) 988 First Protein in explicit water (Levitt PNAS ) From 995 Protein-DNA Complexes Membrane Proteins- Complex Systems

8 Bond Stretching Energy 2 kbn bn b0 n.. bond 2 Unique kb and b0 assigned for each bond pair, i.e. C-C, O-H kb is the spring constant of the bond b0 is the bond length at equilibrium

9 Bond Stretching Force rij V bond V bond rij fi kb rij b0 ri rij ri rij f j f i If atom i and j are closer than b0, the bond force separates them If atom i and j are farther than b0, the bond force draws them nearer

10 Bending Energy 2 k n n 0n angle 2 k is the spring constant of the bending 0 is the angle bending at equilibrium Unique parameters for angle bending are assigned to each bonded triplet of atoms based on their types (e.g. C-C-C, C-O-C, C-CH, etc.)

11 Torsional o Dihedral Energy k cos m dihedral n n n n = angle = phase m = number of peaks in a full rotation

Used for tetrahedral or planar groups 2 k n n 0 n improper 2 dihedral Again

12 Improper Dihedral Energy The energy required to deform a group of atoms from its equilibrium angle, xo. Used for tetrahedral or planar groups 2 k n n 0 n improper 2 dihedral Again this system can be modeled by a spring, and the energy is given by the Hookean potential with respect to the planar angle

13 The Hookean potential kb and k broaden or steepen the slope of the parabola The larger the value of k, the more energy is required to deform an angle (or bond) from its equilibrium value

14 Lennard Jones (Van der Waals) interactions Sir John Lennard Jones Johannes Diderik Van der Waals LJ interactions Cij 2 Cij rij rij nonbonded pairs ij non bonded interactions qi q j + 4 r 0 r ij

15 Electrostatic interactions qi and qj are the partial atomic charges for atoms i and j, separated by a distance rij r is the relative dielectric constant Electrostatic Cij 2 Cij rij rij nonbonded pairs ij non bonded interactions qi q j + 4 r 0 r ij Charles Augustin de Coulomb

Periodic boundary conditions and cut-off radius To simulate our finite system in liquid conditions, we apply the pbc: i.e. the system box is virtually surrounded in all directions by copy of itself.

16 Periodic boundary conditions and cut-off radius To simulate our finite system in liquid conditions, we apply the pbc: i.e. the system box is virtually surrounded in all directions by copy of itself. An atom close to a box border interacts with the atoms in another pbc image. The non-bonded interactions are only calculated between atom pairs closer than a spherical cut-off

17 BOX dimension The edge of cubic box must be large enough to avoid interactions of the solute with itself. Its minimal dimension therefore depends by the chosen cut-off for the non bonded interactions cut-off Edge of the box = 3.2 nm (2.4 nm of water plus the solute size)

Electrostatic interactions: Particle Mesh Ewald (PME) The cut-off radius method for electrostatic interactions is particularly inaccurate for charged molecules such as DNA of for dipolar groups such

18 Electrostatic interactions: Particle Mesh Ewald (PME) The cut-off radius method for electrostatic interactions is particularly inaccurate for charged molecules such as DNA of for dipolar groups such as alpha helices - Short range in the real space - Long range in the Fourier space PME corrects these errors and it helps maintaining short the cut-off in the real space: i.e. the number of atom pairs is reduced Darden et al., Particle mesh Ewald-an NlogN method for Ewald sums in large systems. J. Chem. Phys, 98 (993), 0089

19 Charge groups and atom types A charge group is a neutral charge group composed by several partially charged atoms of a chemical group. Electrostatics can be calculated between charge groups instead that atom pairs Charge groups were first introduced to reduce artifacts in the electrostatics calculation but they can also speed up the calculations; given a pair of water molecules for instance, we only need to determine one atom distance instead of nine (or sixteen for a four-site water model) Note that an atom type is not a physical feature. O8 is defined with a different atom type than O9. In fact, their bond constants with C7 and atomic charges are different

20 Force field V r, r2,..., rn 2 2 k b b k bn n 0n n n 0n bond 2 angle 2 2 k k n cos mn n n n n 0n improper 2 dihedral dihedral Cij 2 Cij rij rij nonbonded pairs ij qi q j + 4 r 0 r ij The potential energy function, together with the parameters required to describe the behavior of different kinds of atoms and bonds (kb, k, k, Cij, ), is called a force field. Several force fields are currently used and the choice depends from the studied system. Some force field are better suited for nucleic acids, for example, while others for membrane proteins

21 Available forcefield in Gromacs (4.6.5). AMBER03 protein, nucleic AMBER94 (Duan et al., J. Comp. Chem. 24, , 2003) 2. AMBER94 force field (Cornell et al., JACS 7, , 995) 3. AMBER96 protein, nucleic AMBER94 (Kollman et al., Acc. Chem. Res. 29, , 996) 4. AMBER99 protein, nucleic AMBER94 (Wang et al., J. Comp. Chem. 2, , 2000) 5. AMBER99SB protein, nucleic AMBER94 (Hornak et al., Proteins 65, , 2006) 6. AMBER99SB-ILDN protein, nucleic AMBER94 (Lindorff-Larsen et al., Proteins 78, , 200) 7. AMBERGS force field (Garcia & Sanbonmatsu, PNAS 99, , 2002) 8. CHARMM27 all-atom force field (with CMAP) - version GROMOS96 43a force field 0. GROMOS96 43a2 force field (improved alkane dihedrals). GROMOS96 45a3 force field (Schuler JCC ) 2. GROMOS96 53a5 force field (JCC 2004 vol 25 pag 656) 3. GROMOS96 53a6 force field (JCC 2004 vol 25 pag 656) 4. GROMOS96 54a7 force field (Eur. Biophys. J. (20), 40,, ) 5. OPLS-AA/L all-atom force field (200 aminoacid dihedrals) 6. [DEPRECATED] Encad all-atom force field, using full solvent charges 7. [DEPRECATED] Encad all-atom force field, using scaled-down vacuum charges 8. [DEPRECATED] Gromacs force field (see manual) 9. [DEPRECATED] Gromacs force field with hydrogens for NMR 2

22 Integration of the equation of motion Numeric integration of Newton's equation of motion is typically done step by step using methods that are called Finite Difference methods. These methods use the information available at time t to predict the system's coordinates and velocities at a time t + t, where t is a short time interval and are based on a Taylor expansion of the position at time t + t r (t t ) r (t ) v(t ) t a (t ) t

23 Integration of the equation of motion r (t t ) 2r (t ) r (t t ) a (t ) t 2 Verlet integrator r (t t ) r (t ) v(t t ) t 2 Leap-frog integrator v(t t ) v(t t ) a(t ) t 2 2 r (t t ) r (t ) v(t ) t a (t ) t 2 Velocity 2 t Verlet v(t t ) v(t ) a (t ) a (t t ) 2

24 Choice of the timestep The length of the timestep must be small compared to the period of the highest frequency motions being simulated Force characteristics Relaxation time (fs) Time step (fs) High frequency motion: bond stretching vibrations Medium frequency motion: angle bending, proper and improper dihedral angle deformation, LJ and short range Coulombian interactions Low frequency motion: long range coulombian interactions The bond stretching vibrations are generally of minimal interest in the study of biomolecular structure and function. Therefore this degree of freedom is usually kept frozen with constraint algorithms (2 x 0-5 s)

25 Constraints The application of geometrical constraints to maintain fixed all the bond lengths, during a simulation, allows the use of a time step up to 2 fs In SHAKE (the first constraints algorithm to be implemented in a MD code) changes a set of unconstrained coordinates to a set of coordinates that fulfill a list of distance constraints, solving a set of Lagrange multipliers in the constrained equations of motion (iterative)

26 Constraints The SETTLE algorithm (developed in 992) is an analytical solution of SHAKE, specifically for water (non-iterative) The LINCS algorithm (developed in 997, twenty years after SHAKE) solves the bond length constraints, always in two steps (non-iterative)

27 Constraints

28 Timescale Protein Folding milliseconds/seconds (0-3-s) Ligand Binding micro/milliseconds ( s) Enzyme catalysis micro/milliseconds ( s) Conformational transitions pico/nanoseconds ( s) Collective vibrations picosecond (0-2 s) Bond vibrations femtosecond (0-5 s)

29 Topology The topology file describes the atoms composing a molecule and their bond connections Es: flexspc.itp in gromacs [ moleculetype ] ; molname nrexcl SOL 2 [ atoms ] ; id at type OW_spc 2 HW_spc 3 HW_spc res nr res name at name cg nr charge mass SOL OW SOL HW SOL HW [ bonds ] ;i j funct length force.c [ angles ] ;i j k funct angle force.c

30 Constraints in Topology Only in case of water, the constraint algorithm can be selected in the topology file flexspc.itp spc.itp

31 Topology

32 Initial velocities The initial velocity of each atom is random assigned through a Maxwell-Boltzmann distribution that is function of the temperature

33 To recapitulate.. Coordinates Velocities Force field Topolog y MD Trajectories Input parameters Analysis

35 Molecular Dynamics ensembles The method discussed above is appropriate for the microcanonical ensemble: constant N (number of particles), V (volume) and ET (total energy = E + Ekin) Note that if time step is short enough, the system loses/gains no net energy (potential + kinetic) when running MD in the NVE ensemble When simulating biological macromolecules, it might be more appropriate to simulate under constant Temperature (T) or constant Pressure (P): Canonical ensemble: NVT Isothermal-isobaric: NPT

36 Simulating at constant T: the Berendsen scheme system Heat bath Bath supplies or removes heat from the system as appropriate dt t T0 T t dt T where determines how strong the bath influences the system Exponentially scale the velocities at each time step by the factor : t T0 T T t 2 T: kinetic temperature Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 8:3684 (984)

37 Simulating at constant T: the Berendsen scheme system Heat bath A small, close to the timestep (strong thermostat), is useful in the equilibration phase, when the quick decreasing of the potential energy could increase too much the kinetic energy of the protein A bigger, e.g. equal to ten times the timestep (weak thermostat), is useful in the production phase, when we want to keep at minimum the perturbation to the conformational sampling Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 8:3684 (984)

38 Simulating at constant P: the Berendsen scheme system Pressure bath dp t P0 P t dt P A change in pressure P is related to a change in volume V Couple the system to a pressure bath To regulate pressure: exponentially scale the volume of the simulation box at each time step by a factor t Po P t where kt : isothermal compressibility t kt P P : coupling constant Berendsen et al. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 8:3684 (984) 3

39 Sample input file of gromacs title = Yo cpp = /lib/cpp include = -I../top define = integrator = md dt = nsteps = nstxout = 5000 nstvout = 5000 nstlog = 5000 nstenergy = 250 nstxout-compressed = 250 compressed-x-grps = Protein energygrps = Protein SOL nstlist = 0 ns-type = grid rlist = 0.8 coulombtype = cut-off rcoulomb =.4 rvdw = 0.8 tcoupl = Berendsen tc-grps = Protein SOL tau-t = ref-t = Pcoupl = Berendsen tau-p =.0 compressibility = 4.5e-5 ref-p =.0 gen-vel = yes gen-temp = 300 gen-seed = constraints = all-bonds

40 Conformational sampling

41 Energy minimization The potential energy surface of a molecule is defined by only a global minumum and a great number of local minima: i.e. conformations where all the first derivative of the potential energy function with respect to the coordinates are zero and all second derivatives are non-negative

42 The energy minimization algorithms finds the nearest local minimum, i.e. the minimum that can be reached by systematically moving down the steepest local gradient Usually they cannot found the global minumum Two algorithms very used in MD codes are steepest descent conjugate gradient Both of the first order: they use the first derivative of the energy potential function with respect to the coordinates

43 The steepest descent uses only the gradient of the potential energy function in the local position to calculate the coordinate movement It is quicker in the single iteration but less precise in finding the local minumum. Therefore is useful in the first steps of minimizations The conjugate gradient uses also the gradient of the potential energy function in the previous step It is more accurate in finding the local minumum but it is slower than the steepest descent. Therefore is usually applied after some steps of steepest descent

44 Starting from A, the steepest descents goes through A-B-C (or A-D-F) While the conjugate gradient, weighting the A-B and B-C gradients, goes through A-B-O

45 GROMACS USER MANUAL

46 -4 interactions Atoms covalently bound are defined as first neighbours second neighbours and so on. LJ and electrostatic interactions are not calculated among first and second neighbours since they are considered in the stretching (first) or in the bending potential (second) The standard non-bonding interactions are too strong for the third neighbours and are reduced (interactions -4; list -4)

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